Seismic data interpolation using nonlinear shaping regularization

Faster nonlinear shaping regularization

Using the definition of equation 9, we define a new shaping operator as:

$$\mathbf{S}'=\mathbf{L}(\mathbf{S}[\mathbf{d}'_n],\mathbf{S}[\mathbf{d}'_{n-1}]),$$ (12)

where $\mathbf{S}'$ is a new version of the commonly defined $\mathbf{S}$ shown in equation 6 and $\mathbf{L}(\cdot,\cdot)$ denotes a linear combination operator. This new shaping operation apply a biased combination between the current model and the previous model, thus is thought to be faster.

Substituting $\mathbf{S}$ in equation 6 with $\mathbf{S}'$ in equation 12, and combined with equation 10, we get a faster version of shaping regularization:

$$\mathbf{d}_{n+1} = \mathbf{L}(\mathbf{S}[\mathbf{d}'_n],\mathbf{S}[\mathbf{d}'_{n-1}]).$$ (13)

The linear combination operator $\mathbf{L}(\mathbf{a},\mathbf{b})$ can be defined as

$$\mathbf{L}(\mathbf{a},\mathbf{b})=\alpha\mathbf{a}+\beta\mathbf{b},$$ (14)

where $\alpha+\beta=1$ .

Seismic data interpolation using nonlinear shaping regularization